Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q ≡ 1 \pmod 3

2024/04/29 by Peter Beelen, Beelen, Peter, Maria Montanucci +3 · 1 citation
Computer Science · Mathematics · #14H05 #14H37 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2404.18808

openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we continue the work started in arXiv:2303.00376v1, explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known \mathbbF_q²-maximal function field Y₃ having the third largest genus, for q ≡ 1 \pmod 3. This function field arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, Y₃ has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of \mathbbF_q²-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, Aut(Y₃) is exactly the automorphism group inherited from the Hermitian function field, apart from small values of q.

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• The automorphism groups of a maximal function field with the second largest genus and its algebraic geometry codes

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